Assuming your non-gaussian noise is independent, adding up enough noise realizations will, through the central limit theorem make them normally distributed.
Good thing, also, is that LDPC codes are linear block codes – meaning that if you add two code words, you are guaranteed to get another code word
So, as strangely as that might sound: instead of LDPC-decoding a single (noisy) codeword, sum up (for example) 10 codewords, and and decode the result. This will not yield a useful infoword, but if all you need is some kind of decision-feedback loop involving successful decoding, then that might be a way.
Another classical way of getting the noise to be normally distributed is to employ a dense, unitarian Matrix as precoder on the transmit side, and its inverse (transpose) on the receive side: You're not changing the number of symbols that way, but you're "combining" enough noise realizations to gaussianizing the noise, whilst not introducing selectivity.
A popular (and quite logical) choice for a matrix pair here is the DFT matrix, because you can very quickly calculate the result using the FFT (and you end up implementing OFDM that way, which can have further interesting properties); you'll definitely want a whitener around that, though, because "nonlinear system" implies "OFDM is kind of a worst case, due to high PAPR", so at least make these worst-case symbols rare.
Unless you can make productive use of the properties of OFDM (e.g. for equalization purposes), using the Walsh-Hadamard transform would be a better choice: easy to compute (using the fast Walsh-Hadamard transform, FWHT), exceptionally well-conditioned, and should be as good as the DFT at gaussianizing your noise.